Connect to multiple VPN with Cisco Client

Your Cisco VPN configuration requires an integrated firewall by policy; that's where the mismatch error is coming from. Problem is Server 2003 only has a very different firewall from 2003 and cisco doesn't think it meets its requirements. You can have the firewall admin (if its not you) disable the firewall policy requirement or you can attempt to enable the built-in firewall through the security configuration wizard in 2003 outlined here -> http://technet.microsoft.com/en-us/library/cc784874%28WS.10%29.aspx

Chances are though, you'll need to have the firewall admin disable the firewall requirement on the vpn configuration from the vpn server as the vpn client may not be able to detect the built-in 2003 server firewall.

The answer to the question title I would say is a pretty hearty 'no' - however...

any way I could get connected with or without the official Sonicwall client

Yes! Certainly this isn't limited to SonicWall, but here's a link to their browser-based VPN appliance. I could've sworn someone was pitching a separate license/service we could purchase for our existing firewall, but I may have misunderstood - I'm not finding a link. However, that would get around your non-admin access to the computers within reach.

You will not be able to install the VPN client through any clever tricks. The VPN adapter functions to windows similarly to a hardware network adapter - that is to say, you need permissions such that you'd be able to add "hardware" to the system. Blocking non-admins from this kind of access is a central part of the distinction between admin and user access.